Causal LTI system: homogeneous-, impulse- and step-response

**1. The problem**

A causal LTI system is described by the difference equation .

Determine the

**(a)** **homogeneous** ( for all ),

**(b)** **impulse**,

**(c)** **step**

**response** of the system.

**2. Relevant equations and definitions**

**2.1** *Causal* are systems for which the output depends only on the input samples , for

**2.2** For the **(a)** part of the problem:

Equation is called the *homogeneous difference equation* and the homogeneous solution. The sequence is in fact a member of a family of solutions of the form

where the coefficients can be chosen to satisfy a set of auxiliary conditions on .

**3. The attempt at a solution**

First of all, there is no additional information to be gained from knowing that the LTI system is also "causal", right? That is, being causal ( only depends on , ) is already in the given equation, correct?

**(a)** **homogeneous response, y_h[n]**

The official solution is given in the form of Eq. :

I have no idea how one should arrive at this solution. Neither writing it out for particular 's (and hoping to glimpse a pattern), nor writing the system in the general form of and then finding the relevant coefficients (N=2: ; M=1: ), gives way toward the solution.

**(b)** **impulse response, h[n]**

I do get the official solution ( ), but I'm exploiting the Z-transform, which is introduced only after this chapter ... so there must be other way without Z-transform:

**(c)** **step response, s[n]**

As in the **(a)** part, I'm completely stuck here. The solution should be , which smells of Z-transform, but I'm also interested if/how could one do without it?

All I can think of for **(c)** is using the result from **(b)** (the impulse response), but I get stuck:

Re: Causal LTI system: homogeneous-, impulse- and step-response

Bumping in hope of an answer.

Re: Causal LTI system: homogeneous-, impulse- and step-response

Think I've found the solution to **(a)**, but I'm still stuck with **(c)**.

**(a)** Since every complex number can be represented as exponential , all that is needed for a homogeneous response, is to solve the characteristic polynomial . Hence, is of the form .

**(c)** Can somebody help me out? Is there an easy way to finding the step response?

Re: Causal LTI system: homogeneous-, impulse- and step-response

**(c)** The solution (the step response) in t-domain via evaluating the convolution sum:

Perhaps simpler to go directly in the z-domain:

Thanks for the help. :)